Finite energy traveling waves for the Gross-Pitaevskii equation in the subsonic regime
Abstract
Ministry of Education, Universities and Research (MIUR)
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FINITE ENERGY TRAVELING WAVES FOR THE GROSS-PITAEVSKII EQUATION IN THE SUBSONIC REGIME JACOPO BELLAZZINI AND DAVID RUIZ Abstract. In this paper we study the existence of finite energy traveling waves for the Gross-Pitaevskii equation. This problem has deserved a lot of attention in the literature, but the existence of solutions in the whole subsonic range was a standing open problem till the work of Mari¸s in 2013. However, such result is valid only in dimension 3 and higher. In this paper we first prove the existence of finite energy traveling waves for almost every value of the speed in the subsonic range. Our argument works identically well in dimensions 2 and 3. With this result in hand, a compactness argument could fill the range of admissible speeds. We are able to do so in dimension 3, recovering the aforementioned result by Mari¸s. The planar case turns out to be more intricate and the compactness argument works only under an additional assumption on the vortex set of the approximating solutions. 1. Introduction In this paper we are concerned with the Gross-Pitaevskii equation (1.1) i∂tΨ = ∆Ψ + Ψ 1−|Ψ|2on Rd×R when d= 2 or d= 3. Observe that this is no more than a Nonlinear Schr¨odinger Equation with a Ginzburg-Landau potential. The Gross-Pitaevskii equation was proposed in 1961 ([31, 47]) to model a quantum system of bosons in a Bose-Einstein condensate, via a Hartree-Fock approximation (see also [3, 6, 36, 37]). It appears also in other contexts such as the study of dark solitons in nonlinear optics ([39, 40]). From the point of view of the dynamics, the Cauchy problem for the Gross-Pitaevskii equation was first studied in one space dimension by Zhidkov [51] and in dimension d= 2,3 by B´ethuel and Saut [13]. At least formally, equation (1.1) presents two invariants, namely: •Energy: E=ZRd 1 2|∇Ψ|2+1 41−|Ψ|22, J.B. was partially supported by “Problemi stazionari e di evoluzione nelle equazioni di campo nonlineari dispersive” of GNAMPA 2020 and the project “Dinamica di equazioni non-lineari dispersive” by FONDAZIONE DI SARDEGNA 2016. D. R. has been supported by the FEDER-MINECO Grant PGC2018-096422-B-I00, by J. Andalucia (FQM-116), and by the Spanish Ministry of Science and Innovation (MICINN), through the IMAGMaria de Maeztu Excellence Grant CEX2020-001105-M/AEI/10.13039/501100011033. 1
2 JACOPO BELLAZZINI AND DAVID RUIZ •Momentum: P=1 2ZRdhi∇Ψ,Ψi, where hf, gi=Re(f)Re(g) + Im(f)Im(g). See also [18, 24, 25, 38] and the references therein for more information on the dynamics of the Gross-Pitaevskii equation. This paper is focused on the existence of traveling wave solutions to (1.1), that is, solutions in the form (1.2) Ψ(x, t) = ψ(x1−ct, ˜x),˜x= (x2. . . xd)∈Rd−1, where the parameter c∈Rcharacterizes the speed of the traveling wave. Without any lack of generality we will consider c > 0 throughout the paper. By the ansatz (1.2) the equation for the profile ψis given by (1.3) ic ∂x1ψ+ ∆ψ+1−|ψ|2ψ= 0. The study of finite energy traveling waves for (1.1) has also implications in the dinamics of the equation. In particular, their pressence is an obstruction to scattering of solutions. Scattering of small energy solutions has been proved in [32, 33] for d= 3, and such result is not true in dimension d= 2. This latter fact may seem surprising for a defocusing Schr¨odinger equation; the reason is that finite energy solutions of (1.1) do not vanish at infinity. Nontrivial finite energy traveling waves in dimension d= 1 are explicitly known, and they are uniquely given (up to rotation or translation) by the expression ψc(x) = r2−c2 2tanh √2−c2 2x!+ic √2, if c < √2. In the literature the function ψ0is called black soliton whereas ψc(c6= 0) receives the name of dark soliton. Their orbital and asymptotic stability has been studied, see [10, 11]. The problem of finding solutions to (1.3) in dimension d≥2 has a long story. In the pioneer work of Jones, Putterman and Roberts ([36, 37]), formal calculations and numerical analysis gave rise to a set of conjectures regarding existence, asymptotic behavior and stability of finite energy travelling waves: the so-called the JonesPutterman-Roberts program. In particular, the existence of finite energy traveling waves is expected if and only if c∈(0,√2) (the sub-sonic case). The threshold value c=√2 comes from the linearization of the problem around the constant solutions of modulus 1. In a certain sense, those solutions correspond to local minima if c < √2. In the last years much progress has been made to give rigorous proofs of those conjectures. Nontrivial finite energy traveling waves for supersonic speed c > √2 do not exist, see [26]. In dimension d= 2 this nonexistence result holds also for c=√2, see [30]. For general nonlinearities analogous results have been proved in [46]. Concerning the asymptotics of finite energy solutions, for any d≥2, finite energy solutions of (1.3) converge at infinity to a fixed complex number of modulus 1. By the
TRAVELING WAVES FOR GROSS-PITAEVSKII EQUATION 3 phase invariance of the problem, we can assume that (1.4) ψ(x)→1 as |x| → +∞. A more precise asymptotic description of ψis indeed available, see [27, 28, 29]. A very active field of research is the study of the location and dynamics of vortices, namely, the zeroes of the wave function ψ. The existence of multi-vortices traveling waves with small speed has been proved in dimension d= 2, see [15, 16, 43]. The existence of solutions with speeds close to √2 has been recently addressed in [44]. In dimension 3 there are traveling vortex rings ([2, 42]) as well as leapfrogging vortex rings, see [35]. At least formally, the Lagrangian associated to (1.3) is defined as: (1.5) Ic(ψ) = E(ψ)−cP(ψ) = 1 2ZRd|∇ψ|2−cP(ψ) + 1 4ZRd1−|ψ|22, where Pis the first component of the momentum Pthat, under suitable integrability conditions (and taken into account (1.4)) can be written as: (1.6) P(ψ) := −ZRd ∂x1(ImΨ)(ReΨ−1). A classical approach to prove existence of traveling waves (starting from [36, 37]) is a minimization procedure of the energy functional Eunder the constraint P(ψ) = pin a suitable functional space. This approach has been pursued in a number of papers, see for instance [8, 12] for the Gross-Pitaevskii equation and [17] for more general nonlinearities. A major difficulty in this strategy is to find a natural definition of the momentum for functions with finite energy, since the integrand in 1.6 might be nonintegrable (see [12]). This approach has the advantage of providing orbital stability of the solutions found (more precisely, of the set of minimizers). As a drawback, the speed cappears as a Lagrange multiplier and is not under control. In particular the possibility of gaps in the subsonic range of velocity cannot be excluded with the constrained minimization approach (see [8]). We shall also quote existence results for small values of c, see [13] in dimension 2 and [14] in dimension 3, but a complete existence result in the sub-sonic case remained for many years as a standing open problem. Finally, Mari¸s proved in [45] the existence result for any c∈(0,√2) in dimension d≥3. His approach is, summing up, to minimize Ic(ψ) under a Pohozaev-type constraint. Once this is accomplished, Mari¸s proves that the corresponding Lagrange multiplier is 0, concluding the proof. This approach works also for more general nonlinearities with nonvanishing conditions at infinity, such as the cubic-quintic nonlinearity. As commented in [45], this minimization approach breaks down in dimension 2 because of different scaling properties: the infimum is 0 and is never attained. One important tool in Mari¸s’ argument is the use of the fiber t7→ ut, where ut(x1,˜x) = u(x1, t˜x). For instance, in dimension d≥4 all solutions correspond to a maximizer of Icwith respect to that fiber. In dimension 3, Ic(ut) is independent of tfor any solution: the argument needs to be adapted, but still the use of the fiber is essential. Those cases have an analogy in the study of the Nonlinear Sch¨odinger
4 JACOPO BELLAZZINI AND DAVID RUIZ equation, see [5], [4], respectively. However, in dimension 2 this approach breaks down, and the fiber utseems of no use; Ic(ut) attains a minimum at t= 1 for any solution u. One of the main motivations of this paper is to deal with the physically relevant 2D model where the existence of finite energy traveling waves in the full subsonic range is still an open problem. Our main result is the following: Theorem 1.1. There exists a subset E⊂(0,√2) of plein measure such that, for any c∈E, there exists a nontrivial finite energy solution of (1.3) ψcsuch that: (1) For any c0∈(0,√2) there exists χ=χ(c0)>0such that 0< Ic(ψc)≤χfor all c∈E, c ≥c0; (2) ind(ψc)≤1, where ind(ψc)stands for the Morse index of ψc, that is, sup{dim Y :Y⊂C∞ 0(Rd)vector space, (Ic)00(ψc)(φ, φ)<0∀φ∈Y} ≤ 1. The proof deals directly with the Lagrangian Icand is focused on searching critical points by using min-max arguments. Our proofs use several ingredients: •Several regularization (or relaxation) techniques have been used in the literature to deal with the Gross-Pitaevskii equation ([13, 45]). Alternatively, some authors have proposed an approach by approximating domains, like flat tori, see [8, 12]. In this paper we choose the second approach, but we use as approximating domains the slabs: (1.7) ΩN=n(x1,˜x)∈R×Rd−1,−N < x1< No, N ∈N. In other words, we first use a mountain-pass argument to address the question of existence of solutions to the problem: (1.8) ic∂x1ψ+ ∆ψ+1−|ψ|2ψ= 0 on ΩN, ψ= 1 on ∂ΩN. The boundary condition is motivated by (1.4). This approach has several advantages. First, as ΩNis bounded in the x1direction, Poincar´e inequality holds and we can work on the space 1 + H1 0(ΩN). As a consequence the momentum given by formula (1.6) is well defined. Secondly, as ΩNis invariant along the variable ˜x, a Pohozaev type inequality is satisfied without boundary terms (see Lemma 2.3). This allows us to avoid the problem of unfolding choices of tori, as in [8, 12]. •A second fundamental tool is an energy bound argument via monotonicity in order to control the energy of (PS) sequences for almost all values of c. This idea has been used many times in literature starting from [49]. The main point here is that we are able to obtain a uniform bound on the energy for a subsequence of enlarging slabs Ωk(N). This is based in a key analytic argument, and it is fundamental in what follows. To the best of our knowledge, this abstract argument is completely new and could be of use in other frameworks where a monotonicity argument is used together with a relaxation procedure.
TRAVELING WAVES FOR GROSS-PITAEVSKII EQUATION 5 •The next step is to pass to the limit, and for that we need to deal with the problem of vanishing. Here we rely on arguments of [8], and we use in an essential way that ψNare solutions of (1.8). We can also exclude the concentration of solutions near the boundary of ΩN, since the problem posed in the half-space ic∂x1ψ+ ∆ψ+1−|ψ|2ψ= 0 on Rd +, ψ= 1 on ∂Rd +, does not admit nontrivial solutions. •Finally, we use the arguments of [20] to obtain a Morse index bound of the solutions obtained. Roughly speaking, since our solutions come from a mountain pass argument, their Morse index is at most 1. This will be used in an essential way in the proof of Theorem 1.3. With Theorem 1.1 in hand, one could ask whether we can pass to the limit and obtain a nontrivial solution for all values of c∈(0,√2). This is relatively easy, see Proposition 6.1. The problem here is to show that the limit solution has finite energy. Let us point out that the boundedness of the energy cannot be deduced only by using Pohozaev-type identities, and more delicate arguments are needed. We give two results on this aspect. The only requirement of the next theorem is d= 3: Theorem 1.2. Assume that d= 3. Let c∈(0,√2),cn∈E,cn→c, where Eis the set given by Theorem 1.1. Let ψnbe the finite energy solutions with speed cngiven by that theorem. Then there exists ξn∈Rdsuch that: ψn(·−ξn)→ψin Ck loc(Rd), where ψis a nontrivial finite energy solution of (1.3) with speed c. Observe that Theorems 1.1 and 1.2 give an alternative proof of the result of Mari¸s [45] for the Gross-Pitaevskii equation. Under minor changes, Theorems 1.1 and 1.2 can be adapted to d≥4: the problem there is the fact that the term (1−|ψ|2)2becomes critical or supercritical with respect to the Sobolev embedding. However, since this term has a positive sign in the functional, this issue could be fixed by changing suitably the functional setting, or, alternatively, by using a convenient truncation argument. For the sake of brevity we will not do so and restrict ourselves to the relevant spatial dimensions d= 2 or d= 3. Regarding compactness of solutions, the case d= 2 is, again, more involved. It presents analytical difficulties and also topological obstructions, see Remarks 6.4, 6.8. In dimension 2 we are able to conclude only under some assumptions on the vortex set of the solutions: Theorem 1.3. Take c∈(0,√2),cn∈Ewith cn→cand ψnthe finite energy solutions with speed cngiven by Theorem 1.1. Assume that (1) either ψnare vortexless, that is, ψn(x)6= 0 for all x∈Rd, (2) or there exists R > 0,δ > 0such that:
6 JACOPO BELLAZZINI AND DAVID RUIZ (1.9) {x∈Rd:ψn(x)=0} ⊂ B(0, R)and |ψn(x)| ≥ δ∀x∈∂B(0, R). Then there exists ξn∈Rdsuch that: ψn(·−ξn)→ψin Ck loc(Rd), where ψis a nontrivial finite energy solution of (1.3) with speed c. The proofs of both Theorem 1.2 and Theorem 1.3 follow similar ideas, which include the following: •A fundamental tool is the use of a lifting, that is, the existence of real functions ρn(x), θn(x) such that ψn=ρneiθn. This is always possible if the solutions are vortexless. If the solutions present vortices, one needs some information on the location of the vortex set. In Theorem 1.2 one can show that the vortices are included in a set of disjoint balls, and that the number of balls and their radius is bounded. Generally speaking, a nonvanishing function ψadmits a lifting if its domain is simply connected. Since the complement of a disjoint union of closed balls is simply connected if d= 3, we can find a lifting outside those balls. In dimension 2 this is no longer true, though, and we can use a lifting only in the complement of one ball, since the total degree of a finite energy solution is 0 (see [27]). •We reason by contradiction assuming that E(ψn)→+∞. A Pohozaev-type identity implies that d X k=2 ZRd|∂xkψn|2= (d−1)Icn(ψn), and Icn(ψn) is bounded by Theorem 1.1. In our arguments we can pass to a limit (locally) which is a 1-D solution of the Gross-Pitaevskii equation (with finite or infinite energy). The knowledge of those 1-D solutions is essential at this point. For instance, in the proof of Theorem 1.3 we are able to obtain in the limit a circular solution ψ(x1) = ρ0eiω0x1, with ρ2 0<2 3(1 + c2/4). But it turns out that such solution has infinite Morse index, and we reach a contradicion. Under minor changes, it is possible to adapt the results of this paper to an equation with more general nonlinearities, namely: ic∂x1ψ+ ∆ψ+F(|ψ|)ψ= 0 on Rd. Several assumptions on the nonlinearity Fwould be in order. However, for the sake of brevity and clarity, we have preferred to focus on the prototype model of the Gross-Pitaevskii equation in this paper. The rest of the paper is organized as follows. Section 2 is devoted to the setting of the notation and some preliminary results. In Section 3 we begin the proof of Theorem 1.1 by considering problem (1.8) from a variational point of view. A main issue here is that we are not able to show that (PS) sequences have bounded energy. This problem is solved for almost all values of cvia the monotonicity trick of Struwe in Section 4.
TRAVELING WAVES FOR GROSS-PITAEVSKII EQUATION 7 We are able to find sequences of slabs Ωk(N)for which those solutions have uniformly bounded energy. In Section 5 we pass to the limit avoiding vanishing or concentration on the boundary, concluding the proof of Theorem 1.1. Sections 6 and 7 are devoted to the proofs of Theorems 1.2, 1.3, respectively. The appendix deals with the Morse index computation of the 1-D circular solutions of the Gross-Pitaevskii equation, which is needed in the conclusion of Theorem 1.3. Acknowledgements: The authors wish to thank Rafael Ortega for many discussions on the 1-D solutions of the Gross-Pitaevskii equation, and also for his help in the elaboration of the Appendix. They also thank the anonymous referee for their careful reading of the manuscript and the many observations that has improved the presentation of our results. 2. Preliminaries In this section we collect some well-known properties of solutions of the GrossPitaevskii equation. We begin by stablishing the notation that we will use throughout the paper. Notation: We denote by hz1, z2ithe real scalar product of two elements in C, that is, hz1, z2i=Re(z1z2). We denote instead by ξ1·ξ2the real scalar product in Rd, to avoid confusion. We shall use the letter ψfor complex valued functions, and we will denote its real and imaginary part by uand v, respectively, so that ψ=u+iv. Moreover, we will write ρto denote its modulus, that is, ρ2=u2+v2=hψ, ψi. We denote the partial derivatives by ∂x1ψ, but sometimes we will use ψx1for convenience. In next lemma we are concerned with the regularity of solutions and the uniform boundedness of their derivatives. Lemma 2.1. Any solution ψof (1.3) or (1.8) is of class C∞and, for any k∈N, there exists Ck>0such that |Dkψ(x)| ≤ Ckfor any x∈Rd. The above result is well-known. The starting point is the L∞estimate: kψkL∞≤p1 + c2/4. This was proved in [22] for all entire solutions of (1.3) (not only those with finite energy). The argument works equally well for problem (1.8) since the boundary condition is compatible with the L∞bound. From this, one can obtain the result via local elliptic regularity estimates. Indeed the solutions are analytic, see [8][Theorem 2.1] for more details. Next lemma gives a Pohozaev identity: Lemma 2.2. Let ψbe a finite energy solution of (1.3). Then: d−2 2ZRd|∇ψ|2−(d−1)cP(ψ) + d 4ZRd1−|ψ|22= 0.
8 JACOPO BELLAZZINI AND DAVID RUIZ Proof. See for instance [26], or [8, Lemma 2.5 and following]. Next identity is also of Pohozaev-type, but only uses the invariance of the domain by dilations in the ˜xvariable: Lemma 2.3. Let ψbe a finite energy solution of either (1.3) or (1.8). Then the following identity holds: (d−3)A(ψ)+(d−1)B(ψ)=0, where A(ψ) = 1 2 d X j=2 Z|∂xjψ|2 and B(ψ) = 1 2Z|∂x1ψ|2+1 4Z1−|ψ|22−cP(ψ). Moreover, by the definition of the Lagrangian (1.5), we conclude that (2.1) Ic(ψ) = 2 d−1A(ψ)≥0. Finally, Ic(ψ)=0if and only if ψis a constant function of modulus 1. Proof. The case of (1.3) has actually been proved in [26][Proposition 5], taking into account [8][Lemma 2.5] (see also [46, Proposition 4.1] or Section 4 in [45]). The case of the domain ΩNis completely analogous and is based on the fact that the dilations (x1,˜x)7→ (x1, λ˜x) leave the domain ΩNinvariant. The following decay estimate has been proved in [27]: Lemma 2.4. Let ψbe a finite energy solution of (1.3) satisfying (1.4). Then the following asymptotics hold: |v(x)| ≤ K 1 + |x|d−1,|u(x)−1| ≤ K 1 + |x|d, |∇v(x)| ≤ K 1 + |x|d,|∇u(x)| ≤ K 1 + |x|d+1 . Outside a ball B(0, R)containing all vortices, ψcan be lifted as ψ=ρeiθ. Then the above decay estimates can be written as: |θ(x)| ≤ K 1 + |x|d−1,|ρ(x)−1| ≤ K 1 + |x|d, |∇θ(x)| ≤ K 1 + |x|d,|∇ρ(x)| ≤ K 1 + |x|d+1 . In particular, the definition (1.6) of the momentum is well defined for any finite energy solution of (1.3). We now define the Morse index of a solution of (1.3):
TRAVELING WAVES FOR GROSS-PITAEVSKII EQUATION 9 Definition 2.5. Let ψbe a solution of (1.3) (either with finite or infinite energy). We define its Morse index ind(ψ)as: sup{dim Y :Y⊂C∞ 0(Rd)vector space, Q(φ)<0∀φ∈Y}, where (2.2) Q(φ) = ZRd|∇φ|2−chφ, i∂x1φi−(1 −|ψ|2)|φ|2+ 2hφ, ψi2. If that set is not bounded from above, we will say that its Morse index is +∞. Observe that, at least formally, Q(φ)=(Ic)00(ψ)[φ, φ], and hence the Morse index is nothing but the maximal dimension for which (Ic)00(ψ)is negative definite. Remark 2.6. An useful property of the so-defined Morse index is that it is decreasing under convergence in compact sets. Being more specific, assume that ψnis a sequence of solutions of (1.3) or (1.8). Assume also that ind(ψn)≤mand ψnconverges to ψ0 in C1 loc sense. Then ind(ψ0)≤m. This property will be essential, in particular, in the proof of Theorem 1.3. Indeed, assume that ind(ψ0)> m; this implies that there exists E⊂C∞ 0(Rd)with dim E =m+ 1 and such that ψ0is negative definite in E. If ψnconverges to ψ0in C1 loc sense, by compactness we obtain that ψnis also negative definite in Efor large n, and hence ind(ψn)> m. 3. The variational approach of Problem (1.8) We first recall the definition of ΩN(1.7) and observe that in the Sobolev Space H1 0(ΩN) the Poincar´e inequality holds: (3.1) ZΩN|φ|2≤CNZΩN|∇φ|2∀φ∈H1 0(ΩN). If we combine this with the Sobolev inequality we obtain that (3.2) kφkLp≤CNk∇φkL2, p∈[2,6] if d= 3, p≥2 if d= 2. Let us define the action functional Ic Nas the Lagrangian Icdefined in (1.5) restricted to the affine space 1 + H1 0(ΩN), that is, Ic N(ψ) := E(ψ)−cP(ψ) = 1 2ZΩN|∇ψ|2−cP(ψ) + 1 4ZΩN1−|ψ|22, with P(u+iv) = −ZΩN (u(x)−1)∂x1v(x). Observe that, integrating by parts, we obtain: (Ic N)0(ψ)(φ) = ZΩNh∇ψ, ∇φi−chi∂x1ψ, φi−(1 −|ψ|2)hψ, φi,
16 JACOPO BELLAZZINI AND DAVID RUIZ (3) Ic k(N)(ψN)≤γk(N)(c). (4) ind(ψN)≤1. One of the key points here is that in (2) the energy is bounded uniformly in N. This will be essential later when passing to the limit as N→+∞. In a first subsection we will give an abstract result, which is basically well-known but maybe not in this specific form. Later we will apply that result to prove Proposition 4.1. 4.1. Energy and Morse index bounds. Energy bounds on Palais-Smale sequences via monotonicity (also called monotonicity trick argument), is a tool first devised in [49] that has been used many times since then, applied to a wide variety of problems. Here we need to adapt this argument to obtain uniform bounds in N, for a subsequence k(N). Moreover, we will also use Morse index bounds for Palais-Smale sequences, in the spirit of [20, 21]. For the sake of completeness, we state and give a proof of a general result in this subsection. Proposition 4.2. Let Xbe a Banach space and A,B:X→Rtwo C1functionals. Assume that either A(ψ)≥0or B(ψ)≥0for all ψ∈X. For any c∈J⊂R+ 0, we define Ic:X→R, Ic(ψ) = A(ψ)−cB(ψ). We assume that there are two points ψ0, ψ1in X, such that setting Γ = {g∈C([0,1], X), g(0) = ψ0, g(1) = ψ1}, the following strict inequality holds for all c∈J: γ(c) = inf g∈Γmax t∈[0,1] Ic(g(t)) >max{Ic(ψ0), Ic(ψ1)}. Then the following assertions hold true: (1) If B≥0,γis decreasing. If instead A≥0, then the map σ(c) = γ(c) cis decreasing. As a consequence, both the maps γ, σ are almost everywhere differentiable. (2) Let c∈J,c > 0, be a point of differentiability of γ. Then, there exists a sequence {ψn}such that (a) Ic(ψn)→γ(c), (b) (Ic)0(ψn)→0in X−1, and (c) dist(ψn, Gn)→0, where Gn={ψ∈X:B(ψ)≤ −γ0(c)+1/n, A(ψ)≤γ(c)−γ0(c)c+1 n}. (3) Let us define, for any δ > 0, the sets (4.1) Fδ={ψ∈X:|Ic(ψ)−γ(c)|<2δ}, Gδ={ψ∈X:B(ψ)<−γ0(c) + δ, A(ψ)<−c2σ0(c) + δ}, Hδ={ψ∈Fδ:dist(ψ, Gδ)<2δ}.
TRAVELING WAVES FOR GROSS-PITAEVSKII EQUATION 17 Let us assume that Aand Bare uniformly C2,α functionals in Hδfor some δ > 0. Then in (2) we can choose ψnsatisfying also that: d) There exists a sequence δn<0,δn→0, such that sup{dim Y :Y⊂Xlinear subspace : (Ic)00(ψn)(φ, φ)≤δnkφk2∀φ∈Y} ≤ 1. Remark 4.3. Observe that, in general, there exist (PS) sequences for Icfor any c∈J; see for instance [1, 50]. The above proposition shows that, for almost all values c∈J, there exist (PS) sequences for Icthat satisfy also condition c). This extra condition c) can be useful in order to show convergence of the (PS) sequence. For instance, if either Aor Bis coercive, Proposition 4.2 implies the existence of bounded (PS) sequences, which is an important information in order to derive convergence. This is the result of [34]. Assertion (3) comes from [20, 21] and gives also a Morse index bound of the (PS) sequence. The only novelty is that we have assumed uniform C2,α regularity on the set Hδ. If Aor Bis coercive, it suffices to have uniform C2,α estimates on bounded sets. Proof. The proof of (1) is inmediate. Indeed, if B≥0, Ic(u) is decreasing in c. Since the family Γ is independent of c, we have that γis decreasing. Instead, if A≥0, then the expression Ic(u) cis decreasing in c, and we conclude. In any of the two cases, the maps γ,σare differentiable in a set E⊂Jof plein measure. In order to prove (2), we are largely inspired by [34]. We first state and prove the following lemma: Lemma 4.4. Let c∈E,c > 0, then there exists gn∈Γsuch that (1) maxt∈[0,1] Ic(gn(t)) →γc. (2) There exists ρn>0,ρn→0such that for all t∈[0,1] with Ic(gn(t)) ≥γ(c)−1 n, we have: B(gn(t)) ≤ −γ0(c) + ρn, A(gn(t)) ≤ −c2σ0(c) + ρn. Proof of the lemma. Take cn∈Jan increasing sequence converging to c. For any n∈N, there exists gn∈Γ such that maxt∈[0,1] Icn(gn(t)) ≤γ(cn) + |cn−c|2. If B≥0 we have that: max t∈[0,1] Ic(gn(t)) ≤max t∈[0,1] Icn(gn(t)) ≤γ(cn) + |cn−c|2→γ(c). Instead, if A≥0, max t∈[0,1] Ic(gn(t)) ≤c cn max t∈[0,1] Icn(gn(t)) ≤c cn (γ(cn) + |cn−c|2)→γ(c). We now take t∈[0,1] such that Ic(gn(t)) ≥γ(c)−|c−cn|2. Then: B(gn(t)) = Icn(gn(t)) −Ic(gn(t)) c−cn ≤γ(cn) + |cn−c|2−γ(c) + |cn−c|2 c−cn→ −γ0(c).
18 JACOPO BELLAZZINI AND DAVID RUIZ Moreover, lim sup n→+∞A(gn(t)) = lim sup n→+∞Ic(gn(t)) + cB(gn(t)) ≤γ(c)−cγ0(c). It suffices then to take cn=c−1 √n. Recall now the definitions of Fδ,Gδand Hδgiven in (4.1). By the previous lemma the set Fδ∩Gδis not empty: indeed, the curves gnpass through Fδ∩Gδfor sufficiently large n. Proposition 4.2, (2) is proved if we show that for any δ > 0, inf{k(Ic)0(ψ)k:ψ∈Hδ}= 0. We argue by contradiction, and assume that there exists δ > 0 such that inf{k(Ic)0(ψ)k: ψ∈Hδ} ≥ δ > 0. A classical deformation argument shows that there exists ε > 0, η∈C([0,1] ×X:X) such that: i) η(s, ψ) = ψif s= 0, |Ic(ψ)−γ(c)|>2εor dist(ψ, Gδ)>2δ. ii) Ic(η(1, ψ)) ≤γ(c)−εfor all ψ∈Gδwith Ic(ψ)≤γ(c) + ε. iii) η(s, ·) is a homeomorphism of X. iv) kη(s, ψ)−ψk< δ, v) Ic(η(s, ψ)) ≤Ic(ψ) for all ψ∈X. The existence of the above deformation can be found in [50, Lemma 2.3], for instance. Actually our notation is compatible with that reference, setting S=Gδ, and taking ε=δ2/8, for instance. We now take nlarge enough and the curve gngiven by the lemma. If Ic(gn(t)) < γ(c)−1 n, by b), we have that Ic(η(1, gn(t))) < γ(c)−1 n. In on the contrary, Ic(gn(t)) ≥ γ(c)−1 n, we can combine the lemma with ii) to conclude that Ic(η(1, gn(t))) ≤γ(c)−ε. As a consequence, max tIc(η◦gn(t)) < γ(c), a contradiction. For the proof of (3) of Proposition 4.2, we use Theorem 1 of [20] to our sequence of paths gn. It is important to point out that, in our setting, the uniform C2,α regularity assumption in [20] is needed only in the set Hδdefined above. Indeed, the proof of Theorem 1.bis of [20] (pages 93-94) only needs a uniform C2,α bound, independent of n, in a certain ball B(gn(t),2¯ δ), where |Ic(gn(t)) −γ(c)| ≤ ¯ε, for some ¯ δ > 0, ¯ε > 0. Observe that by Lemma 4.4, such ball is contained in Hδ, with a suitable choice of the constants. Finally, since by [20] d(ψn, gn[0,1]) →0 and Ic(ψn)→γ(c), again Lemma 4.4 implies that ψn∈Hδfor large n. By a diagonal argument, we can take ψnsuch that d(ψn, Gn)→0.
TRAVELING WAVES FOR GROSS-PITAEVSKII EQUATION 19 4.2. Proof of Proposition 4.1. A direct application of the above results to our setting, combined with Proposition 3.3, yields the existence of finite energy solutions in any domain ΩN, for almost all values of c. The problem here is that the energy of those solutions could diverge if we make N→+∞. In order to obtain uniform bounds independent of the parameter N, we need a more subtle application of Proposition 4.2. Define: (1) X= 1 + H1 0(ΩN), which is an affine Banach space, for which Proposition 4.2 also holds; (2) A(ψ) = E(ψ), which is positive and coercive; (3) B(ψ) = P(ψ), the momentum; (4) J= (c0,√2) for a fixed value c0>0. For N≥N0the functional Ic Nhas a min-max geometry (see Proposition 3.1); recall that γN(c)>0 is the function that associates to a speed c∈Jthe min-max value of Ic N. Clearly, σN(c) = γN(c) cis decreasing in cas Proposition 4.2 shows. By Proposition 4.2, there exists a bounded (PS) sequence in H1 0(ΩN) at level γN(c). Proposition 3.3 yields then the existence of a solution ψNwith: Ic N(ψN)≤γN(c),E(ψN) = A(ψN)≤ −c2σ0 N(c). Since Ais coercive, here the set Hδis uniformly bounded, and Iis clearly uniformly C2,α in bounded sets. By Proposition 4.2, 3), we have that: sup{dim Y :Y⊂H1 0(ΩN):(Ic N)00(ψN)(φ, φ)<0∀φ∈Y} ≤ 1. We are now concerned with passing to the limit as N→+∞. In order to control the energy of the solutions ψN, we reason as follows. Recall Proposition 3.1, b), and that σN(c) is decreasing in c; then, for N≥N0, (4.2) χ(c0) c0≥γN(c0) c0≥γN(c0) c0−γN(c) c≥Zc c0|σ0 N(s)|ds. Let us now define the sets DN,M ={c∈(c0,√2) : σNis not differentiable or |σ0 N(c)|> M}, for all N,M∈N,N≥N0. Clearly the sets DN,M also depend on c0, but we avoid to make that dependence explicit in the notation for the sake of clarity. By (4.2), we have that |DN,M | ≤ χ(c0) c0M. The following claim is the key to be able to pass to the limit for enlarging slabs preserving bounded energy. Claim: The set D(c0) defined as: D(c0) = ∩M∈N∪N≥N0∩k≥NDk,M has 0 measure.
20 JACOPO BELLAZZINI AND DAVID RUIZ Indeed, the sets ∩k≥NDk,M are increasing in N, and all of them satisfy that have measure smaller than χ(c0) c0M. Hence the same estimate works also for the union in N. Now, D(c0) is a set given by an intersection of sets of measure χ(c0) c0M,M∈N, so that D(c0) has 0 measure. Finally, we can set D=∪+∞ n=1 D(1/n), which has also 0 measure. Let us define E= (0,√2) \D, and take c∈E. We can fix n∈Nsuch that c0= 1/n < c, and c /∈D(c0). Then, there exists M(c) and a subsequence k(N) such that |σ0 k(N)(c)| ≤ M(c). By Proposition 4.2, for any of these slabs Ωk(N)there exists a Palais-Smale sequence with bounded energy. According to Proposition 3.3, this gives rise to a solution ψk(N)∈1 + H1 0(Ωk(N)) such that: Ic k(N)(ψk(N))≤γk(N)(c),E(ψk(N))≤M(c)c2. This concludes the proof of Proposition 4.1. 5. Proof of Theorem 1.1 In view of Proposition 4.1, we aim to conclude the proof of Theorem 1.1 by passing to the limit. This is indeed possible thanks to Lemma 2.1. However, we need to face two difficulties: vanishing of solutions (that is, the limit solution is trivial) and concentration near the boundary (that is, the limit solution is defined in a half-space). The purpose of this section is to exclude both scenarios. Next result deals with the question of vanishing and is actually a version of Proposition 2.4 of [8] adapted to problem (1.8). Proposition 5.1. Let ψbe a nontrivial finite energy solution of (1.8) with 0< c < √2, then k1−|ψ|kL∞(ΩN)≥2 5(1 −c √2). The proof is actually the same as in [8], with one difference: when integrating by parts, the authors use the decay estimates of the solutions to avoid contributions from infinity, and those estimates are available only for the Euclidean space case. Instead, here we use integrability bounds. In our argument we will use liftings of the solutions, that is, we write ψ=ρeiθ. The existence of liftings is always guaranteed, for instance, if |ψ(x)| 6= 0 for all x. 5.1. Liftings for solutions in ΩNwithout vortices. We consider here solutions without vortices, i.e. that do not vanish. The energy density is given by the following formula e(ρ, θ) = 1 2|∇ρ|2+|∇θ|2ρ2+1 41−|ρ|22
TRAVELING WAVES FOR GROSS-PITAEVSKII EQUATION 21 and the associated energy is E(ρ, θ) := ZΩN e(ρ, θ) By using the fact that ψ=ρeiθ is a solution of (1.3), ρ, θ fulfill the following system of equations (5.1) c 2∂x1ρ2+∇·(ρ2∇θ) = 0, cρ∂x1θ−∆ρ−ρ(1 −ρ2) + ρ|∇θ|2= 0. . The following pointwise inequality (Lemma 2.3 in BGS) (5.2) (ρ2−1)∂x1θ≤√2 ρe(ρ, θ) that holds for arbitary C1scalar function that can be written as ψ=ρeiθ (not necessary being a solution) are crucial in the sequel. Lemma 5.2. Let ψbe a vortexless finite energy solution in ΩN, then 1−ρand θbelong to H1 0(ΩN). Proof. Let us notice that for a vortexless finite energy solution (5.3) E(ψ) = 1 2ZΩN|∇ρ|2+ρ2|∇θ|2<+∞ which implies, by means of Poincar´e inequality, that ρ−1∈H1 0(ΩN). Since ∇ρbounded in L∞(by Lemma 2.1), one concludes that ρ(x)→1 uniformly as |x| → +∞. Hence ρ(x)≥ρ0>0 for all x∈ΩN. Again by (5.3), ∇θ∈L2(ΩN). To conclude the proof we shall prove that θ= 0 on ∂ΩNwhich will allows to use Poincar´e inequality. Let us assume that θ= 0 if x1=−Nand that θ= 2πif x1=N. For any ˜x∈Rd−1 there exists y∈(−N, N) such that u(y, ˜x) = 0. We get 1 = |u(y, ˜x)−u(−N, ˜x)|2=Zy −N ∂x1u(s, ˜x)ds 2 ≤2NZN −N|∂x1u(s, ˜x)|2ds. By Fubini we get ZΩN|∂x1u|2=ZRd−1ZN −N|∂x1u(s, ˜x)|2dsd˜x= +∞, which implies that the energy is infinity. From (5.1) we derive three useful identities that are important in the sequel. These identities have been stablished in [8, Lemmas 2.8, 2.10] for solutions in the whole euclidean space: here we adapt these arguments to the problem in the domain ΩN.
22 JACOPO BELLAZZINI AND DAVID RUIZ Lemma 5.3. Let ψbe a vortexless finite energy solution of (1.8). Then: (5.4) P(ψ) = 1 2ZΩN (1 −ρ2)∂x1θ. (5.5) cP=ZΩN ρ2|∇θ|2. (5.6) ZΩN2ρ|∇ρ|2+ρ(1 −ρ2)2=cZΩN ρ(1 −ρ2)∂x1θ+ZΩN ρ(1 −ρ2)|∇θ|2. Proof. Straightforward computation gives P(ψ) = 1 2ZΩN ∂x1(ρsin θ)−ρ2∂x1θ=1 2ZΩN ∂x1(ρsin θ−θ) + (1 −ρ2)∂x1θ. We will prove that RΩN∂x1(ρsin θ−θ) = 0. Thanks to Lemma 5.2 ψ1= (1 −ρ2)∂x1θ is integrable in ΩNand hence we derive that ψ2=∂x1(ρsin θ−θ) is integrable as well. By integration by parts together with Lemma 5.2 we get ZΩN ψ2=Z∂ΩN (ρsin θ−θ)η1= 0. To get (5.5) we multiply the first equation of (5.1) by θand we integrate in ΩN,M , defined as ΩN,M =nx∈Rd,−N < x1< N, |xj|< M, 2≤j≤do⊂ΩN. By integrating by parts we obtain: c 2ZΩN,M (1 −ρ2)∂x1θ−ZΩN,M ρ2|∇θ|2 =Z∂ΩN,M θc 2(1 −ρ2)η1−ρ2∇θ·η. Observe that by Lemma 5.2 all functions involved in the expression above belong to L1(ΩN), and recall that θ= 0 on ∂ΩN. Then, there exists a sequence Mnsuch that lim n→∞Z∂ΩN,Mn θρ2∇θ·η= 0. This proves (5.5). By multiplying the second equation of (5.1) by ρ2−1 and integrating over ΩN,M by parts we obtain ZΩN,M 2ρ|∇ρ|2+ρ(1 −ρ2)2+Z∂ΩN,M (1 −ρ2)∇ρ·η= =cZΩN,M ρ(1 −ρ2)∂x1θ+ZΩN,M ρ(1 −ρ2)|∇θ|2.
TRAVELING WAVES FOR GROSS-PITAEVSKII EQUATION 23 Again by Lemma 5.2, all functions involved in the above expression belong to L1(ΩN). Hence we can finde a sequence Mnsuch that lim n→∞Z∂ΩN,Mn (1 −ρ2)∇ρ·η= 0. This proves (5.6) passing to the limit. 5.2. Proof of Proposition 5.1. Let us call δ=||1−|ψ|||L∞(ΩN). If δ > 1 2>2 5(1−c √2) there is nothing to prove. Let us suppose hence that δ < 1 2which implies that ρ(x)≥ 1−δ > 1 2for any x∈ΩN. In particular ψadmits a lifting ψ=ρeiθ. We notice that 4(1 −δ)ZΩN 1 2|∇ρ|2+1 41−|ρ|22≤ZΩN 2ρ|∇ρ|2+ρ1−|ρ|22 and thanks to (5.6) we get (5.7) ZΩN e(ρ, θ)≤1 4(1 −δ)ZΩN ρ(1 −ρ2)c∂x1θ+|∇θ|2+1 2ZΩN ρ2|∇θ|2 The strategy is to estimate r.h.s of (5.7) using the pointwise bound given by (5.2). We have, thanks to (5.5) and (5.2) c 4(1 −δ)ZΩN ρ(1 −ρ2)∂x1θ+1 2ZΩN ρ2|∇θ|2≤ √2c 4(1 −δ)+√2c 4!ZΩN e(ρ, θ) and hence c 4(1 −δ)ZΩN ρ(1 −ρ2)∂x1θ+1 2ZΩN ρ2|∇θ|2≤c √2(1 −δ)ZΩN e(ρ, θ). Now we claim that (5.8) ZΩN ρ(1 −ρ2)|∇θ|2≤6δZΩN e(ρ, θ) such that we obtain ZΩN e(ρ, θ)≤(c √2(1 −δ)+3δ 2(1 −δ))ZΩN e(ρ, θ). The fact that e(ρ, θ)≥0 and that 1 −(c √2(1−δ)+3δ 2(1−δ))≤0 if δ≥2 5(1−c √2) concludes the proof. Now we prove claim (5.8). Notice that ZΩN ρ(1 −ρ2)|∇θ|2≤δZΩN ρ(1 + ρ)|∇θ|2. Now, ρ(1 + ρ)≤3ρ2if ρ≥1 2, such that thanks to (5.5) ZΩN ρ(1 −ρ2)|∇θ|2≤3δZΩN ρ2|∇θ|2≤3δc 2ZΩN (1 −ρ2)∂x1θ≤3√2δc ZΩN e(ρ, θ). The proof of the claim ends noticing that 0 < c < √2.
24 JACOPO BELLAZZINI AND DAVID RUIZ 5.3. Conclusion of the proof of Theorem 1.1. Take c∈E,c0∈(0, c) and N0 given by Proposition 3.1. By Proposition 5.1, there exists ξNsuch that |ψk(N)(ξN)−1|90, where ψk(N)are the solutions given by Proposition 4.1. We consider now a space translation defining ˜ ψk(N)(x) = ψk(N)(x−ξN). Observe that the uniform bounds of Lemma 2.1 allow us to use Ascoli-Arzel`a Theorem for the sequence ˜ ψk(N), which converges locally to a certain function ψc. If d(ξN, ∂Ωk(N))→+∞, the limit function ψcis a nontrivial solution of (1.3) in Rd. Moreover, by Fatou Lemma, E(ψc) is finite. Finally, again by Fatou lemma and (2.1), I(ψc) = 1 d−1 d X j=2 Z|∂xjψc|2≤1 d−1lim inf N→+∞ d X j=2 Z|∂xj˜ ψk(N)|2 = lim inf N→+∞Ik(N)(ψk(N))≤lim inf N→+∞γk(N)(c)≤χ(c0). This shows the validity of Theorem 1.1 in this case. Assume now that d(ξN, ∂Ωk(N)) is bounded. Up to a subsequence, the limit function ψcis a nontrivial solution of (1.3) defined in a half-space {x∈Rd:x1>−m}or {x∈Rd:x1< m}, for some m > 0, and with boundary condition ψc= 1. Again by Fatou Lemma, E(ψc) is finite. In next proposition we rule out this possibility, and this concludes the proof of Theorem 1.1. Proposition 5.4. Let ψbe a finite energy solution of the problem: (5.9) ic∂x1ψ+ ∆ψ+1−|ψ|2ψ= 0 on Rd +, ψ= 1 on ∂Rd +, where c∈Rand Rd +={x∈Rd:x1>0}.Then ψ= 1. Proof. The proof follows well-known ideas that date back to [19]. If ψis a finite energy solution, then ∇ψand (1 −|ψ|2) are functions in L2(Rd +). Since ψis in L∞(Rd +) and is a strong solution, standard regularity results allow us to conclude that D2ψbelongs to L2(Rd +). Hence we can multiply equation (5.9) by ∂x1ψand integrate by parts, obtaining: cZRd +hi∂x1ψ, ∂x1ψi= 0; ZRd +h∆ψ, ∂x1ψi=Z∂Rd +h(∇ψ·ν), ∂x1ψi−ZRd + 1 2∂x1|∇ψ|2 =−Z∂Rd +|∂x1ψ|2+1 2Z∂Rd +|∂x1ψ|2=−1 2Z∂Rd +|∂x1ψ|2; ZRd +1−|ψ|2hψ, ∂x1ψi=−1 4ZRd + ∂x1(1 −|ψ|2)2= 0. These computations imply that:
TRAVELING WAVES FOR GROSS-PITAEVSKII EQUATION 25 Z∂Rd +|∂x1ψ|2= 0. In other words, ∂x1ψ= 0 in ∂Rd +. By unique continuation, we conclude that ψ= 1. 6. Proof of Theorem 1.2 In this section we prove the compactness criterion given in 1.2. We start by the following result, which is independent of the dimension: Proposition 6.1. Let d= 2 or 3,cn→c,cn∈Ewhere the set Eis given by Theorem 1.1. Let ψnbe the sequence of solutions provided by that theorem. Then there exists ξn∈Rdsuch that ψn(·−ξn)converges locally in Ck(up to a subsequence) to a nontrivial solution ψ0of (1.3). Proof. Let ψbe a nontrivial finite energy solution of (1.3) with 0 < c < √2. Then there exists ε=ε(c)>0 such that (6.1) k1−|ψ|kL∞(Rd)≥ε. Statement (6.1) is just Proposition 2.4 of [8]. Compare it with Proposition 5.1, which is nothing but its version for problem (1.8) (with a slight change of the constants). Then, there exists ξnsuch that |1−|ψn(ξn)|| > ε for some fixed ε > 0. By Lemma 2.1 we can use Ascoli-Arzel`a Theorem to obtain that ψn(·−ξn) converges locally in Ck to a nontrivial solution ψ0of (1.3). The main problem to conclude the proof of Theorem 1.2 or 1.3 is to assure that ψ0has finite energy. Let us point out that the boundedness of the energy cannot be deduced only by using the Pohozaev identities given in Lemmas 2.2, 2.3. Observe that since Icn(ψn) is bounded, Lemma 2.3 implies that 3 X j=2 ZR3|∂xjψn|2=O(1). The idea of the proof is to try to relate the behavior of ψnwith that of the 1-D solutions of the Gross-Pitaevskii equation. Next proposition is a first step in this line (see also Remark 6.3). Proposition 6.2. Let ψnbe solutions of (1.3) for cn,cn→c, such that Icn(ψn)≤C. Then, ZR3|∇gn|2+ZR3|∇hn|2=O(1), where (6.2) gn= (∂x1un)vn−(∂x1vn)un−cn 2(ρ2 n−1),
32 JACOPO BELLAZZINI AND DAVID RUIZ Assume by contradiction that |Sr n|is bounded for some r > c √2. Observe that: Icn(ψn) = ZR3\∪N k=1Bn k l(ρn, θn) + O(1) = Z{ρn≥r} l(ρn, θn) + O(1). We now use the inequality |c 2(1−ρ2 n)∂x1θn| ≤ (1−ρ2)2 4(1+ε)+c2 4(∂x1θn)2(1+ε) with suitable ε > 0 to obtain: Z{ρn≥r} l(ρn, θn)≥Z{ρn≥r} 1 2|∇ρn|2+1 2−c2(1 + ε) 4ρ2 n|∇θn|2ρ2 n+ε 1 + ε (1 −ρ2 n)2 4 ≥ε0Z{ρn≥r} e(ρn, θn) = ε0E(ψn) + O(1), for suitable ε0>0. Then, O(1) = Icn(ψn)≥ε0E(ψn) + O(1), and this allows us to conclude. 6.1. Proof of Theorem 1.2. With all the results above we can inmediately conclude the proof of Theorem 1.2. Indeed, by (6.8) and Sobolev inequality, we have that ZR3 (1 −ρn)6=O(1). If E(ψn)→+∞, Proposition 6.10 implies |Sr n|is unbounded for r > c √2, and this is a contradiction with the above estimate. Hence E(ψn) is bounded. By Fatou Lemma, the solution ψ0given in Proposition 6.1 has finite energy, concluding the proof. 7. Proof of Theorem 1.3 In this section we prove the compactness criterion given in Theorem 1.3. The proof follows some of the ideas of the previous section, but with important differences. As previously, we will be done if we show that E(ψn) is bounded. By (1.9), we have that ψn6= 0 outside B(0, R); as a consequence, ψnadmit a lifting ψn(x) = ρn(x)eiθn(x)for all x∈R2\B(0, R). This is a consequence of the fact that ψn have finite energy, see [27][Lemma 15]. In the vortexless case, this lifting holds in the whole euclidean space. Next lemma is a version of Lemma 6.9: Lemma 7.1. Take c∈(0,√2),cn∈Ewith cn→cand ψnthe solutions given by Theorem 1.1. Assume also that there exists R > 0and δ > 0such that (1.9) is satisfied. Then (7.1) P(ψn) = 1 2ZB(0,R)c (1 −ρ2 n)∂x1θn+O(1), (7.2) cP(ψn) = ZB(0,R)c ρ2 n|∇θn|2+O(1),
TRAVELING WAVES FOR GROSS-PITAEVSKII EQUATION 33 (7.3) ZB(0,R)c|∇ρn|2=O(1). Proof. The proof is completely analogue to that of Lemma 6.9. Observe that in the vortexless case we have exact identities in (7.1), (7.2), (7.3). With Lemma 6.9 in hand, we can adapt the proof of Proposition 6.10 to our setting, obtaining the following result: Proposition 7.2. Take c∈(0,√2),cn∈Ewith cn→cand ψnthe solutions given by Theorem 1.1. Assume that: E(ψn)→+∞. Then, for any r∈(c √2,1),|Sr n| → +∞. Next result is analogue to Lemma 6.6. The only difference is that now we do not know that (6.5) holds, but instead we have (7.3). Lemma 7.3. Under the assumptions of Theorem 1.3, assume that for some r∈(0,1), |Sr n| → +∞. Then, there exists ξn∈Sr nand Rn→+∞such that: ZB(ξn,Rn)|∇ρn|2+|∂x2ψn|2→0. Proof. The proof is analogue to that of Lemma 6.6. 7.1. Proof of Theorem 1.3. Assume by contradiction that E(ψn)→+∞. By Proposition 7.2, we can apply Lemma 7.3 to a value rsatisfying that: c √2< r < r2 3(1 + c2/4) <1. Notice that this is possible if c < √2. Let ξn∈Rdgiven by Lemma 7.3 and define ˜ ψn(x) = ψn(x−ξn). Up to a subsequence we have that: ˜ ψn→ψ0in Ck loc(Rd). Taking into account Remark 2.6, ind(ψ0)≤1. By Lemma 6.6, ψ0depends only of the x1variable. Moreover ∇ρ0= 0 where ρ0=|ψ0| ≤ r. That is, ψ0(x1) is a 1D circular solution, ψ0(x1) = ρ0eiω(x1−t), where ω2+cω +ρ2 0= 1. By the choice of r, we have that ρ2 0<2 3(1 + c2/4). But those solutions have infinite Morse index, as shown in Proposition 8.1 (see Appendix). This contradiction shows that E(ψn) is bounded. By Fatou Lemma, the solution ψ0given in Proposition 6.1 has finite energy, concluding the proof.
34 JACOPO BELLAZZINI AND DAVID RUIZ Remark 7.4. Let us point out that Theorem 1.2 does not need the information on the Morse index of the solutions. The main tool there is that Icn(ψn) = O(1). Instead, Theorem 1.3 requires in a essential way that the Morse index of the solutions obtained is bounded. 8. Appendix (by Rafael Ortega) In this appendix we prove the following result: Proposition 8.1. Given t∈R,ω0∈R,ρ0>0satisfying that ω2 0+cω0+ρ2 0= 1, the function ψ0(x) = ρ0eiω0(x−t)is a (infinite energy) solution of (1.3). Assume also that ρ2 0<2 3(1 + c2/4). Then its Morse index, as defined in Definition 2.5, is infinity. Proof. The problem is autonomous so that we can assume t= 0. The proof is based on the study of the 1Dproblem: (8.1) ψ00 +icψ0+1−|ψ|2ψ= 0 on R. By the change of variables φ=eixc/2ψwe pass to a problem: (8.2) φ00 +1 + c2/4−|φ|2φ= 0 on R. The Morse index of this problem depends on the existence of conjugate points to some solutions of the linearized equation, see for instance [23, Chapter 5]. The function φ(x) = ρ0eiω1xis a solution of (8.2), where , ω1=ω0+c/2. Observe that (8.3) ω2 1+ρ2 0= 1 + c2/4 The linearized equation to (8.2) around the solution φis: ζ00 + (1 + c2/4)ζ−2φ(s)φ(s)ζ−φ(s)2ζ= 0. We will follow the lines of [48, Section 21] to analyze the oscillatory properties of this equation. ζ00 + (1 + c2/4)ζ−2ρ2 0ζ−ρ2 0e2iω1sζ= 0. We now make the change of variable ζ=eiω1sη, to obtain a constant coefficient linear system: (8.4) η00 + 2iω1η0−ρ2 0η−ρ2 0η= 0. If ρ2 0<2ω2 1(which, by (8.3), reduces to ρ2 0<2 3(1 + c2/4)) we can find the explicit solution to (8.4): η(s) = sin sp4ω2 1−2ρ2 0 p4ω2 1−2ρ2 0 +iω1 cos sp4ω2 1−2ρ2 0−1 2ω2 1−ρ2 0 . Clearly, ζ(s) = eiω1sη(s) has infinitely many conjugate points 2πn √4ω2 1−2ρ2 0 ,n∈N.
TRAVELING WAVES FOR GROSS-PITAEVSKII EQUATION 35 Given any interval I, the quadratic functional ˜ Q1,I :H1 0(I, C)→R, ˜ Q1,I(σk) = ZI|σ0 k|2−(1 + c2/4−|φ|2)|σk|2+ 2(hσk, φi)2<0 is in the conditions of Section 29.2 of [23]. We can apply [23, Theorem 3’ in page 122] to deduce that ˜ Q1,I takes negative values as soon as the length of the interval Iis greater than 2π √4ω2 1−2ρ2 0 . Then we can find infinitely many functions σk∈C∞ 0(R)with disjoint support such that ˜ Q1(σk) = Z∞ −∞ |σ0 k|2−(1 + c2/4−|φ|2)|σk|2+ 2(hσk, φi)2<0. We now want to pass to the original problem (8.1) and estimate its Morse index. In order to do so, define τk(s) by σk(s) = eics/2τk(s). Simple computations give: σ0 k(s)=(ic 2τk(s) + τ0 k(s))eics/2, |σ0 k(s)|2=|τ0 k(s)|2+c2 4|τk(s)|2+chiτk(s), τ0 k(s)i=|τ0 k(s)|2+c2 4|τk(s)|2−chτk(s), iτ0 k(s)i. Moreover, hσk(s), φ(s)i=hτk(s), ψ(s)i. As a consequence ˜ Q1(σk) = Q1(τk)<0, where Q1(τk) = Z∞ −∞ |τ0 k|2−chτk, iτ0 ki−(1 −|ψ|2)|τk|2+ 2(hτk, ψi)2. Observe that this is the quadratic form associated to (8.1). Take now a C∞ 0function χk:Rd−1→R+, and let us estimate Qon the function ιk(x) = χk(˜x)τk(x1), where Qis defined in (2.2): Q(ιk) = Q1(τk)ZRd−1 χk(˜x)2d˜x+Z+∞ −∞ |τk(x1)|2dx1ZRd−1|∇χk(˜x)|2d˜x. It suffices to take now χksuch that RRd−1χ2 k= 1 and RRd−1|∇χk|2is sufficiently small, to conclude that Q(ιk)<0. Observe also that supp ιk∩supp ιk0=∅if k6=k0, since an analogue property holds for σkand τk. Hence, Qis negative definite on the vector space generated by the linearly independent functions {ι1, . . . , ιk}for any k∈N, concluding the proof.
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38 JACOPO BELLAZZINI AND DAVID RUIZ [50] M. Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhuser Boston, Inc., Boston, MA, 1996. [51] P.E. Zhidkov, The Cauchy problem for the nonlinear Schr¨odinger equation, Joint Inst. Nucl. Res., Dubna (1987), 15 pp. J. Bellazzini, Dipartimento di Matematica, Universit` a Degli Studi di Pisa, Largo Bruno Pontecorvo, 5, 56127, Pisa, Italy E-mail address:[email protected] David Ruiz IMAG, Universidad de Granada, Departamento de An´ alisis Matem´ atico, Campus Fuentenueva, 18071 Granada, Spain E-mail address:[email protected]